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1. CJM Online first

Rotger, Victor; de Vera-Piquero, Carlos
 Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves The purpose of this note is introducing a method for proving the existence of no rational points on a coarse moduli space $X$ of abelian varieties over a given number field $K$, in cases where the moduli problem is not fine and points in $X(K)$ may not be represented by an abelian variety (with additional structure) admitting a model over the field $K$. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired on the work of Ellenberg and Skinner on the modularity of $\mathbb{Q}$-curves, is that to a point $P=[A]\in X(K)$ represented by an abelian variety $A/\bar K$ one may still attach a Galois representation of $\operatorname{Gal}(\bar K/K)$ with values in the quotient group $\operatorname{GL}(T_\ell(A))/\operatorname{Aut}(A)$, provided $\operatorname{Aut}(A)$ lies in the centre of $\operatorname{GL}(T_\ell(A))$. We exemplify our method in the cases where $X$ is a Shimura curve over an imaginary quadratic field or an Atkin-Lehner quotient over $\mathbb{Q}$. Keywords:Shimura curves, rational points, Galois representations, Hasse principle, Brauer-Manin obstructionCategories:11G18, 14G35, 14G05

2. CJM Online first

Vandenbergen, Nicolas
 On the Global Structure of Special Cycles on Unitary Shimura Varieties In this paper, we study the reduced loci of special cycles on local models of the Shimura variety for $\operatorname{GU}(1,n-1)$. Those special cycles are defined by Kudla and Rapoport. We explicitly compute the irreducible components of the reduced locus of a single special cycle, as well as of an arbitrary intersection of special cycles, and their intersection behaviour in terms of Bruhat-Tits theory. Furthermore, as an application of our results, we prove the connectedness of arbitrary intersections of special cycles, as conjectured by Kudla and Rapoport. Keywords:Shimura varieties, local models, special cyclesCategory:14G35

3. CJM Online first

Abdesselam, Abdelmalek; Chipalkatti, Jaydeep
 On Hilbert Covariants Let $F$ denote a binary form of order $d$ over the complex numbers. If $r$ is a divisor of $d$, then the Hilbert covariant $\mathcal{H}_{r,d}(F)$ vanishes exactly when $F$ is the perfect power of an order $r$ form. In geometric terms, the coefficients of $\mathcal{H}$ give defining equations for the image variety $X$ of an embedding $\mathbf{P}^r \hookrightarrow \mathbf{P}^d$. In this paper we describe a new construction of the Hilbert covariant; and simultaneously situate it into a wider class of covariants called the GÃ¶ttingen covariants, all of which vanish on $X$. We prove that the ideal generated by the coefficients of $\mathcal{H}$ defines $X$ as a scheme. Finally, we exhibit a generalisation of the GÃ¶ttingen covariants to $n$-ary forms using the classical Clebsch transfer principle. Keywords:binary forms, covariants, $SL_2$-representationsCategories:14L30, 13A50

4. CJM Online first

Guardo, Elena; Harbourne, Brian; Van Tuyl, Adam
 Symbolic Powers Versus Regular Powers of Ideals of General Points in $\mathbb{P}^1 \times \mathbb{P}^1$ Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci-Harbourne developed methods to address this problem, which involve asymptotic numerical characters of symbolic powers of the ideals. Most of the work done up to now has been done for ideals defining 0-dimensional subschemes of projective space. Here we focus on certain subschemes given by a union of lines in $\mathbb{P}^3$ which can also be viewed as points in $\mathbb{P}^1 \times \mathbb{P}^1$. We also obtain results on the closely related problem, studied by Hochster and by Li-Swanson, of determining situations for which each symbolic power of an ideal is an ordinary power. Keywords:symbolic powers, multigraded, pointsCategories:13F20, 13A15, 14C20

5. CJM Online first

Goulden, I. P.; Guay-Paquet, Mathieu; Novak, Jonathan
 Monotone Hurwitz Numbers in Genus Zero Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero. Keywords:Hurwitz numbers, matrix models, enumerative geometryCategories:05A15, 14E20, 15B52

6. CJM 2012 (vol 65 pp. 634)

Mezzetti, Emilia; Miró-Roig, Rosa M.; Ottaviani, Giorgio
 Laplace Equations and the Weak Lefschetz Property We prove that $r$ independent homogeneous polynomials of the same degree $d$ become dependent when restricted to any hyperplane if and only if their inverse system parameterizes a variety whose $(d-1)$-osculating spaces have dimension smaller than expected. This gives an equivalence between an algebraic notion (called Weak Lefschetz Property) and a differential geometric notion, concerning varieties which satisfy certain Laplace equations. In the toric case, some relevant examples are classified and as byproduct we provide counterexamples to Ilardi's conjecture. Keywords:osculating space, weak Lefschetz property, Laplace equations, toric threefoldCategories:13E10, 14M25, 14N05, 14N15, 53A20

7. CJM 2012 (vol 65 pp. 575)

 The Geometry and Fundamental Group of Permutation Products and Fat Diagonals Permutation products and their various fat diagonal'' subspaces are studied from the topological and geometric point of view. We describe in detail the stabilizer and orbit stratifications related to the permutation action, producing a sharp upper bound for its depth and then paying particular attention to the geometry of the diagonal stratum. We write down an expression for the fundamental group of any permutation product of a connected space $X$ having the homotopy type of a CW complex in terms of $\pi_1(X)$ and $H_1(X;\mathbb{Z})$. We then prove that the fundamental group of the configuration space of $n$-points on $X$, of which multiplicities do not exceed $n/2$, coincides with $H_1(X;\mathbb{Z})$. Further results consist in giving conditions for when fat diagonal subspaces of manifolds can be manifolds again. Various examples and homological calculations are included. Keywords:symmetric products, fundamental group, orbit stratificationCategories:14F35, 57F80

8. CJM Online first

Thompson, Alan
 Explicit models for threefolds fibred by K3 surfaces of degree two We consider threefolds that admit a fibration by K3 surfaces over a nonsingular curve, equipped with a divisorial sheaf that defines a polarisation of degree two on the general fibre. Under certain assumptions on the threefold we show that its relative log canonical model exists and can be explicitly reconstructed from a small set of data determined by the original fibration. Finally we prove a converse to the above statement: under certain assumptions, any such set of data determines a threefold that arises as the relative log canonical model of a threefold admitting a fibration by K3 surfaces of degree two. Keywords:threefold, fibration, K3 surfaceCategories:14J30, 14D06, 14E30, 14J28

9. CJM Online first

Aholt, Chris; Sturmfels, Bernd; Thomas, Rekha
 A Hilbert Scheme in Computer Vision Multiview geometry is the study of two-dimensional images of three-dimensional scenes, a foundational subject in computer vision. We determine a universal GrÃ¶bner basis for the multiview ideal of $n$ generic cameras. As the cameras move, the multiview varieties vary in a family of dimension $11n-15$. This family is the distinguished component of a multigraded Hilbert scheme with a unique Borel-fixed point. We present a combinatorial study of ideals lying on that Hilbert scheme. Keywords:multigraded Hilbert Scheme, computer vision, monomial ideal, Groebner basis, generic initial idealCategories:14N, 14Q, 68

10. CJM 2012 (vol 64 pp. 1222)

Bobiński, Grzegorz
 Normality of Maximal Orbit Closures for Euclidean Quivers Let $\Delta$ be an Euclidean quiver. We prove that the closures of the maximal orbits in the varieties of representations of $\Delta$ are normal and Cohen--Macaulay (even complete intersections). Moreover, we give a generalization of this result for the tame concealed-canonical algebras. Keywords:normal variety, complete intersection, Euclidean quiver, concealed-canonical algebraCategories:16G20, 14L30

11. CJM Online first

Adamus, Janusz; Randriambololona, Serge; Shafikov, Rasul
 Tameness of complex dimension in a real analytic set Given a real analytic set $X$ in a complex manifold and a positive integer $d$, denote by $\mathcal A^d$ the set of points $p$ in $X$ at which there exists a germ of a complex analytic set of dimension $d$ contained in $X$. It is proved that $\mathcal A^d$ is a closed semianalytic subset of $X$. Keywords:complex dimension, finite type, semianalytic set, tamenessCategories:32B10, 32B20, 32C07, 32C25, 32V15, 32V40, 14P15

12. CJM 2012 (vol 65 pp. 544)

Deitmar, Anton; Horozov, Ivan
 Iterated Integrals and Higher Order Invariants We show that higher order invariants of smooth functions can be written as linear combinations of full invariants times iterated integrals. The non-uniqueness of such a presentation is captured in the kernel of the ensuing map from the tensor product. This kernel is computed explicitly. As a consequence, it turns out that higher order invariants are a free module of the algebra of full invariants. Keywords:higher order forms, iterated integralsCategories:14F35, 11F12, 55D35, 58A10

13. CJM 2012 (vol 65 pp. 120)

Francois, Georges; Hampe, Simon
 Universal Families of Rational Tropical Curves We introduce the notion of families of $n$-marked smooth rational tropical curves over smooth tropical varieties and establish a one-to-one correspondence between (equivalence classes of) these families and morphisms from smooth tropical varieties into the moduli space of $n$-marked abstract rational tropical curves $\mathcal{M}_{n}$. Keywords:tropical geometry, universal family, rational curves, moduli spaceCategories:14T05, 14D22

14. CJM 2012 (vol 65 pp. 195)

Penegini, Matteo; Polizzi, Francesco
 Surfaces with $p_g=q=2$, $K^2=6$, and Albanese Map of Degree $2$ We classify minimal surfaces of general type with $p_g=q=2$ and $K^2=6$ whose Albanese map is a generically finite double cover. We show that the corresponding moduli space is the disjoint union of three generically smooth irreducible components $\mathcal{M}_{Ia}$, $\mathcal{M}_{Ib}$, $\mathcal{M}_{II}$ of dimension $4$, $4$, $3$, respectively. Keywords:surface of general type, abelian surface, Albanese mapCategories:14J29, 14J10

15. CJM 2011 (vol 64 pp. 1090)

Rosso, Daniele
 Classic and Mirabolic Robinson-Schensted-Knuth Correspondence for Partial Flags In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson-Schensted-Knuth correspondence. Then we use this result to generalize the mirabolic Robinson-Schensted-Knuth correspondence defined by Travkin, to the case of two partial flags and a line. Keywords:partial flag varieties, RSK correspondenceCategories:14M15, 05A05

16. CJM 2011 (vol 64 pp. 1248)

Gärtner, Jérôme
 Darmon's Points and Quaternionic Shimura Varieties In this paper, we generalize a conjecture due to Darmon and Logan in an adelic setting. We study the relation between our construction and Kudla's works on cycles on orthogonal Shimura varieties. This relation allows us to conjecture a Gross-Kohnen-Zagier theorem for Darmon's points. Keywords:elliptic curves, Stark-Heegner points, quaternionic Shimura varietiesCategories:11G05, 14G35, 11F67, 11G40

17. CJM 2011 (vol 64 pp. 3)

Boissière, Samuel
 Automorphismes naturels de l'espace de Douady de points sur une surface On Ã©tablit quelques rÃ©sultats gÃ©nÃ©raux relatifs Ã  la taille du groupe d'automorphismes de l'espace de Douady de points sur une surface, puis on Ã©tudie quelques propriÃ©tÃ©s des automorphismes provenant d'un automorphisme de la surface, en particulier leur action sur la cohomologie et la classification de leurs points fixes. Keywords:SchÃ©ma de Hilbert, automorphismes, points fixesCategory:14C05

18. CJM 2011 (vol 64 pp. 1122)

 $p$-adic $L$-functions and the Rationality of Darmon Cycles Darmon cycles are a higher weight analogue of Stark--Heegner points. They yield local cohomology classes in the Deligne representation associated with a cuspidal form on $\Gamma _{0}( N)$ of even weight $k_{0}\geq 2$. They are conjectured to be the restriction of global cohomology classes in the Bloch--Kato Selmer group defined over narrow ring class fields attached to a real quadratic field. We show that suitable linear combinations of them obtained by genus characters satisfy these conjectures. We also prove $p$-adic Gross--Zagier type formulas, relating the derivatives of $p$-adic $L$-functions of the weight variable attached to imaginary (resp. real) quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express the second derivative of the Mazur--Kitagawa $p$-adic $L$-function of the weight variable in terms of a global cycle defined over a quadratic extension of $\mathbb{Q}$. Categories:11F67, 14G05

19. CJM 2011 (vol 64 pp. 845)

Helm, David; Katz, Eric
 Monodromy Filtrations and the Topology of Tropical Varieties We study the topology of tropical varieties that arise from a certain natural class of varieties. We use the theory of tropical degenerations to construct a natural, multiplicity-free'' parameterization of $\operatorname{Trop}(X)$ by a topological space $\Gamma_X$ and give a geometric interpretation of the cohomology of $\Gamma_X$ in terms of the action of a monodromy operator on the cohomology of $X$. This gives bounds on the Betti numbers of $\Gamma_X$ in terms of the Betti numbers of $X$ which constrain the topology of $\operatorname{Trop}(X)$. We also obtain a description of the top power of the monodromy operator acting on middle cohomology of $X$ in terms of the volume pairing on $\Gamma_X$. Categories:14T05, 14D06

20. CJM 2011 (vol 64 pp. 805)

Chapon, François; Defosseux, Manon
 Quantum Random Walks and Minors of Hermitian Brownian Motion Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam, and van Moerbeke that the process of eigenvalues of two consecutive minors of a Hermitian Brownian motion is a Markov process; whereas, if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion. Keywords:quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor processCategories:46L53, 60B20, 14L24

21. CJM 2011 (vol 63 pp. 1345)

Jardine, J. F.
 Pointed Torsors This paper gives a characterization of homotopy fibres of inverse image maps on groupoids of torsors that are induced by geometric morphisms, in terms of both pointed torsors and pointed cocycles, suitably defined. Cocycle techniques are used to give a complete description of such fibres, when the underlying geometric morphism is the canonical stalk on the classifying topos of a profinite group $G$. If the torsors in question are defined with respect to a constant group $H$, then the path components of the fibre can be identified with the set of continuous maps from the profinite group $G$ to the group $H$. More generally, when $H$ is not constant, this set of path components is the set of continuous maps from a pro-object in sheaves of groupoids to $H$, which pro-object can be viewed as a Grothendieck fundamental groupoid". Keywords:pointed torsors, pointed cocycles, homotopy fibresCategories:18G50, 14F35, 55B30

22. CJM 2011 (vol 64 pp. 123)

Lee, Jae-Hyouk
 Gosset Polytopes in Picard Groups of del Pezzo Surfaces In this article, we study the correspondence between the geometry of del Pezzo surfaces $S_{r}$ and the geometry of the $r$-dimensional Gosset polytopes $(r-4)_{21}$. We construct Gosset polytopes $(r-4)_{21}$ in $\operatorname{Pic} S_{r}\otimes\mathbb{Q}$ whose vertices are lines, and we identify divisor classes in $\operatorname{Pic} S_{r}$ corresponding to $(a-1)$-simplexes ($a\leq r$), $(r-1)$-simplexes and $(r-1)$-crosspolytopes of the polytope $(r-4)_{21}$. Then we explain how these classes correspond to skew $a$-lines($a\leq r$), exceptional systems, and rulings, respectively. As an application, we work on the monoidal transform for lines to study the local geometry of the polytope $(r-4)_{21}$. And we show that the Gieser transformation and the Bertini transformation induce a symmetry of polytopes $3_{21}$ and $4_{21}$, respectively. Categories:51M20, 14J26, 22E99

23. CJM 2011 (vol 64 pp. 409)

Rainer, Armin
 Lifting Quasianalytic Mappings over Invariants Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of invariant polynomials $\mathbb C[V]^G$. We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$ over the mapping of invariants $\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$ and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass $\mathcal C \subseteq C^\infty$ satisfying some mild closedness properties that guarantee resolution of singularities in $\mathcal C$, e.g., the real analytic class, then $f$ admits a lift of the same class $\mathcal C$ after desingularization by local blow-ups and local power substitutions. As a consequence we show that $f$ itself allows for a lift that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation. If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions. Keywords:lifting over invariants, reductive group representation, quasianalytic mappings, desingularization, bounded variationCategories:14L24, 14L30, 20G20, 22E45

24. CJM 2011 (vol 64 pp. 81)

David, C.; Wu, J.
 Pseudoprime Reductions of Elliptic Curves Let $E$ be an elliptic curve over $\mathbb Q$ without complex multiplication, and for each prime $p$ of good reduction, let $n_E(p) = | E(\mathbb F_p) |$. For any integer $b$, we consider elliptic pseudoprimes to the base $b$. More precisely, let $Q_{E,b}(x)$ be the number of primes $p \leq x$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$, and let $\pi_{E, b}^{\operatorname{pseu}}(x)$ be the number of compositive $n_E(p)$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$ (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address the problem of finding upper bounds for $Q_{E,b}(x)$ and $\pi_{E, b}^{\operatorname{pseu}}(x)$, generalising some of the literature for the classical pseudoprimes to this new setting. Keywords:Rosser-Iwaniec sieve, group order of elliptic curves over finite fields, pseudoprimes Categories:11N36, 14H52

25. CJM 2011 (vol 63 pp. 1058)

Easton, Robert W.
 $S_3$-covers of Schemes We analyze flat $S_3$-covers of schemes, attempting to create structures parallel to those found in the abelian and triple cover theories. We use an initial local analysis as a guide in finding a global description. Keywords:nonabelian groups, permutation group, group covers, schemesCategory:14L30
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